Analytical investigations of in situ stress inversion from borehole breakout geometries

This study aims to investigate the feasibility of deriving in situ horizontal stresses from the breakout width and depth using the analytical method.Twenty-three breakout data with different borehole sizes were collected and three failure criteria were studied.Based on the Kirsch equations,relatively accurate major horizontal stress(sH)estimations from known minor horizontal stress(sh)were achieved with percentage errors ranging from 0.33%to 44.08%using the breakout width.The Mogi-Coulomb failure criterion(average error:13.1%)outperformed modified Wiebols-Cook(average error:19.09%)and modified Lade(average error:18.09%)failure criteria.However,none of the tested constitutive models could yield reasonable sh predictions from known sH using the same approach due to the analytical expression of the redistributed stress and the nature of the constitutive models.In consideration of this issue,the horizontal stress ratio(sH/sh)is suggested as an alternative input,which could estimate both sH and sh with the same level of accuracy.Moreover,the estimation accuracies for both large-scale and laboratory-scale breakouts are comparable,suggesting the applicability of this approach across different breakout sizes.For breakout depth,conformal mapping and complex variable method were used to calculate the stress concentration around the breakout tip,allowing the expression of redistributed stresses using binomials composed of sH and sh.Nevertheless,analysis of the breakout depth stabilisation mechanism indicates that additional parameters are required to utilise normalised breakout depth for stress estimation compared to breakout width.These parameters are challenging to obtain,especially under field conditions,meaning utilising normalised breakout depth analytically in practical applications faces significant challenges and remains infeasible at this stage.Nonetheless,the normalised breakout depth should still be considered a critical input for any empirical and statistical stress estimation method given its significant correlation with horizontal stresses.The outcome of this paper is expected to contribute valuable insights into the breakout stabilisation mechanisms and estimation of in situ stress magnitudes based on borehole breakout geometries.

Using Refined Theory to Studied Elastic Wave Scattering and Dynamic Stress Concentrations in Plates with Two Cutouts

In this paper, based on complex variables and conformal mapping methods, using the refined dynamic equation of plates, elastic wave scattering and dynamic stress concentrations in plates with two cutouts were studied. Applying the orthogonal function expansion method, the problem to be solved can be reduced into the solution of a set of infinite algebraic equations. According to free boundary conditions, numerical results of dynamic moment concentration factors in thick plates with two circular cutouts analyze that: there will be more complex interaction changes between two-cutout situation than single cutout situation. In the case of low frequency or high frequency and thin plate, the hole-spacing in the absence of coupling interactions was larger or smaller. The numerical results and method can be used to analyze the dynamics and strength of plate-like structures.

基于复变函数不同规格矩形巷道围岩应力分布规律解析

鉴于矩形巷道应用的广泛性,为深入剖析矩形巷道尺寸、侧压力系数等因素变化对巷道围岩应力的影响程度及规律,将矩形巷道围岩应力求解视为弹性力学孔口问题,采用复变函数理论与Schwartz−Chistoffel映射函数,结合保角变换进行分析得到ζ复平面单位圆ζ的解析函数φ0(ζ)、ψ0(ζ),再进行Laurent级数展开后,求得复势函数φ(ζ)、ψ(ζ)。经过进一步推导最终确定矩形巷道围岩环向应力解析解,在此基础上通过环向应力的极坐标表达,深入剖析了矩形巷道尺寸与侧压力系数变化下矩形巷道周边围岩应力变化规律以及影响程度。结果表明:取映射函数前三项计算,映射巷道断面已逼近于理论断面,可保证精度要求;随着巷道宽高比增大,以宽高比等于1为分界点,巷道围岩周边应力峰值先增大后减小;巷道帮部围岩应力随宽高比的增大逐渐减小,巷道顶底板围岩应力随宽高比的增大逐渐增大;侧压力系数越大,侧帮围岩应力峰值越大,顶底板应力峰值越小;侧帮围岩应力随侧压力的增大而增大,两者呈正相关,顶底板应力随侧压力增大而减小,两者呈负相关。

Anti-plane Analysis of a Circular Hole withThree Unequal Cracks in One-dimensionalHexagonal Piezoelectric Quasicrystals

This paper employes variable function method and the technique of conformal mappingto discuss the anti-plane problem of a circular hole with three unequal cracksin a one-dimensional （1D） hexagonal piezoelectric quasicrystal. Based on the piezoelectricityfundamental equations of quasicrystal materials and the symmetry of1D hexagonal quasicrystal and its linear piezoelectricity effect, 1D hexagonal quasicrystalcontrol equations of anti-plane problem are derived. Applying Cauchyintegral formula, the analytical expressions for the crack tip filed intensity factorsare presented with the assumption that the crack are electrical impermeable andelectrical permeable. With the variation of the hole-size and the crack length, someof the new model of crack are obtained. In the absence of the electric load, theresults match with the classical ones. The numerical results indicate the effects ofgeometric parameters on the field intensity factors. It is verified that the horizontalcrack length and the circle radius can easily promote crack growth. Researchon such issues will provide reliable theoretical value for the engineering materialspreparation and application.

An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks

The existing analytical solutions are extended to obtain the stress fields and the stress intensity factors(SIFs) of two unequal aligned cracks emanating from an elliptical hole in an infinite isotropic plane. A conformal mapping is proposed and combined with the complex variable method. Due to some difficulties in the calculation of the stress function, the mapping function is approximated and simplified via the applications of the series expansion. To validate the obtained solution, several examples are analyzed with the proposed method, the finite element method, etc. In addition, the effects of the lengths of the cracks and the ratio of the semi-axes of the elliptical hole(a/b) on the SIFs are studied. The results show that the present analytical solution is applicable to the SIFs for small cracks.

INTEGRAL EQUATION METHOD'S APPLICATION IN HOLE-EDGE STRESS OF COMPOSITE MATERIAL PLATE WITH DIFFERENT SHAPED HOLES

The strength of composite plate with different hole-shapes is always one of the most important but complicated issues in the application of the composite material. The holes will lead to mutations and discontinuity to the structure. So the hole-edge stress concentration is always a serious phenomenon. And the phenomenon makes the structure strength decrease very quickly to form dangerous weak points. Most partial damage begins from these weak points. According to the complex variable functions theory, the accurate boundary condition of composite plate with different hole-shapes is founded by conformal mapping method to settle the boundary condition problem of complex hole-shapes. Composite plate with commonly hole-shapes in engineering is studied by several complex variable stress fimction. The boundary integral equations are founded based on exact boundary conditions. Then the exact hole-edge stress analytic solution of composite plate with rectangle holes and wing manholes is resolved. Both of offset axis loadings and its influences on the stress concentration coefficient of the hole-edge are discussed. And comparisons of different loads along various offset axis on the hole-edge stress distribution of orthotropic plate with rectangle hole or wing manhole are made. It can be concluded that hole-edge with continuous variable curvatures might help to decrease the stress concentration coefficient; and smaller angle of outer load and fiber can decrease the stress peak value.

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to analyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With the method of complex variable function, a series' of conformal mapping functions are obtained from different cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And, the general expressions for the equations of a cylindrical shell, including the solutions of stress concentrations meeting the boundary conditions of the large openings' edges, are given in the mapping plane. Furthermore, by applying the orthogonal function expansion technique, the problem can be summarized into the solution of infinite algebraic equation series. Finally, numerical results are obtained for stress concentration factors at the cutout's edge with various opening's ratios and different loading conditions. This new method, at the same time, gives a possibility to the research of cylindrical shells with large non-circular openings or with nozzles.

AN ANALYSIS FOR DIFFRACTION OF FLEXURAL WAVES AND DYNAMIC STRESS CONCENTRATIONS IN THICK PLATES WITH A CAVITY

By using the complex variable method and conformal mapping, the diffraction of flexural waves and dynamic stress concentrations in thick plates with a cavity have been studied. A general solution of the stress problem of the thick plate satisfying the boundary conditions on the contour of an arbitrary cavity is obtained. By employing the orthogonal function expansion technique, the dynamic stress problem can be reduced to the solution of an infinite algebraic equation series. As an example, the numerical results for the dynamic stress concentration factor in thick plates with a circular, elliptic cavity are graphically presented. The numerical results are discussed.

INFLUENCE OF COUPLE-STRESSES ON STRESS CONCENTRATIONS AROUND THE CAVITY

The complex function method was used in the solution of micropolar elasticity theory around cavity in an infinite elasticity plane. In complex plane, the general solution of two dimension micropolar elasticity theory is given. The solution comes from analytic function and 'Zonal Function'. The boundary conditions of non-circular cavity are satisfied by using the conformal mapping method. Based on the method, a general approach solving the stress concentration in micropolar elasticity theory is established. Finally, the numerical calculation is carried out to the stress concentration coefficient of circular cavity.

ON THE STRESS CONCENTRATION IN THICK CYLINDRICAL SHELLS WITH AN ARBITRARY CUTOUT

In this paper, based on the theory of thick shells including effects of transverse shear deformations, a complex variables analytic method to solve stress concentrations in circular cylindrical shells with a small cutout is established A general solution and expression satisfying the boundary conditions on the edge of arbitrary cutouts are obtained. The stress problem can be reduced to the solution of an infinite algebraic equation series, and can be normalized by means of this method. Numerical results for stress concentration factors of the shell with a small circular and elliptic cutout are presented.

THE GENERAL SOLUTION FOR THE STRESS PROBLEM OF CIRCULAR CYLINDRICAL SHELLS WITH AN ARBITRARY CUTOUT

In this paper a complex variable analytic method for solving stress concentrations in the circular cylindrical shell is proposed. The problem to be solved can be summarized into the solution of an infinite algebraic equation series. The solution can be normal and effective by means of this method. Numerical results for stress concentrations in the shell with a circular, elliptic cutout are graphically presented.

On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle

In this work, we used the complex variable methods to derive the Goursat functions for the first and second fundamental problem of an infinite plate with a curvilinear hole C. The hole is mapped in the domain inside a unit circle by means of the rational mapping function. Many special cases are discussed and established of these functions. Also, many applications and examples are considered. The results indicate that the infinite plate with a curvilinear hole inside the unit circle is very pronounced.

Optimal thermal design of anisotropic plates with arbitrary cutouts using genetic algorithm

Anisotropic plates in different applications may have geometric defects such as openings and cracks.The presence of the opening disturbs the heat flow,which creates significant thermal stress around the opening.When the heat flux is high enough,these extreme stresses can lead to structural failure.This article aims to obtain the optimal parameters for achieving the minimum value of the normalized stress near the cutout’s boundary in perforated anisotropic plates utilizing the genetic algorithm.Optimization parameters include the curvature of opening’s corners,orientation angle of opening,fibers angle,heat flux angle,and opening’s elongation.The plate is under heat flux,and the opening’s border is thermally insulated.The stress distribution around the opening is calculated using Lekhnitskii’s complex variable method and complex potential functions.The genetic algorithm is then implemented to find the optimal values for design parameters.The results show that by selecting the optimal parameters related to the anisotropic material and the opening’s geometry,the stress intensity factor of the perforated anisotropic plates is remarkably reduced.Furthermore,this optimization algorithm can be extended to find the optimized parameters and achieve the optimal designs in anisotropic and isotropic perforated plates under thermal loadings.

平面弹性波入射下空沟隔振的解析分析

基于复变函数理论和保角映射方法,采用波函数展开法及多级坐标变换技术,解析求解了半无限空间中空沟对平面SH波和平面P波的隔振问题。首先,通过设虚源法和大圆弧假定法分别给出了平面SH波和平面P波入射半无限空间时的散射场。然后由空沟边界及大圆弧边界处的应力自由边界条件建立无穷阶线性方程组,并通过对该方程组的求解给出了空沟对平面SH波和平面P波散射的解析解答。最后,参数分析了激励频率、空沟深度等因素对隔振效果的影响规律。结果表明,空沟对平面SH波和平面P波的隔振效果随着入射波频率的增大而增大,随着空沟深度增大而增大。

亚纯映射分担移动目标的唯一性定理

研究亚纯映射分担移动目标的唯一性问题,在Quang-An所做的关于移动目标的第二基本定理的基础上考虑重值,获得了带截断重数的第二基本定理,并应用所得的第二基本定理获得了两个带截断重数的亚纯映射分担移动目标的唯一性定理,推广了前人所做的工作。

基于改进迭代求解的双孔洞围岩应力场解析研究

针对含双孔洞岩体应力迭代求解计算过程中多余面力的求解难题,提出新的理论解析方法。通过引入最优化技术,直接进行不同坐标系和平面中孔洞边界坐标点的变换,无需在求解过程中引入逆映射函数,避免了逆映射函数求解造成的误差。随后,以双拱形孔洞为例,对不同布局下的孔周应力场进行求解与分析,并利用数值模拟对理论求解结果进行验证。研究结果表明:采用所提出的方法可以精确获得双孔洞周边应力分布,显著降低求解难度;在单轴压缩条件下,孔洞体系的初始萌生裂纹为孔洞顶底板拉应力集中区域形成的拉伸裂纹,但最终破坏表现为双孔洞间的贯通破坏;不同连接角的双孔洞体系峰值应力与压应力集中系数呈线性相关关系,说明双孔洞岩石的破坏取决于孔洞边界的压应力集中程度。

基于复变函数不同形状的巷道围岩应力分析

深埋巷道开挖后围岩应力重新分布,影响巷道稳定。因此分析巷道围岩二次应力分布状态对于评估围岩稳定性或设计支护参数具有重要意义。目前岩石力学相关理论中讨论了圆形巷道、椭圆形巷道和矩形巷道二次应力弹性分布特征,缺乏对六边形巷道的分析。本文从弹性力学的孔口问题为切入点,基于复变函数理论,通过保角变换将矩形、六边形巷道转化为单位圆,结合弹性力学理论建立了矩形、六边形巷道围岩分析的力学模型,并分析得到巷道围岩应力分布的解析解。将解析结果与数模拟分析结果对比可知,相同结构巷道围岩二次应力分布一致且大小基本相同,就应力集中情况而言,六边形巷道优于矩形巷道。

浅埋非圆形隧道开挖引起的力学响应解析解

浅埋非圆形隧道的力学分析对城市地铁隧道的施工安全具有重要意义,其求解难点在于如何考虑重力场的影响以及圆环域保角映射函数的确定.为此,在复变函数理论框架下通过重构解析函数的表达,提出了重力条件下浅埋非圆形隧道力学分析的解耦保角映射方法,该方法将解析函数拆分成两组子函数,并采用不同的局部坐标系进行表达,两组子函数可以分别表达地表边界内与隧道边界外的应力与位移场,同时地表边界与隧道边界也可以单独进行保角映射;利用快速Fourier变换将用于确定解析函数的边界条件方程转化为频域方程进行求解;将本文方法应用于浅埋隧道开挖引起的地表沉降分析.研究结果表明:隧道形状会对地表沉降的大小造成显著影响,其主因素来源于隧道的高跨比;隧道埋深会同时对地表沉降量以及沉降槽宽度产生影响,埋深越小其敏感程度越大;侧压力系数的改变对地表沉降槽的宽度影响较小;与有限元方法相比,本文方法的程序体量极小、计算速度更快且精度更高.

波阻板对平面P-SV波的散射及其隔振性能分析

在二维弹性地基的有限深度处埋设波阻板,采用复弹性理论分析波阻板对平面P-SV波的散射及其隔振效果。首先,通过保角映射技术给出矩形波阻板的映射表达式;其次,根据复弹性理论建立问题在复数域上的动力学控制方程和应力边界条件,并结合波函数展开法得到动力响应解答;最后,引入振幅衰减系数,并通过其变化曲线分析平面P-SV波作用下埋置深度、剪切模量比、弹性波入射角度以及弹性波频率等对隔振效果的影响。结果表明:剪切模量比对波阻板隔振效果影响显著,且随着模量比增大,隔振效果增强;波阻板埋深越大,隔振效果越差;随着入射波角度和频率增大,隔振效果也随之降低。

浅埋隧道围岩应力场的计算复变函数求解法

对于受地表边界和地面荷载影响的浅埋隧道的围岩应力场,由于在数学处理上存在一定的困难,很难用解析解来进行分析,而通常采用边界元或有限元的数值方法来解答。为了求解浅埋隧道的应力场,采用边界配点来确定边界条件,同时用保角映射将一个含圆孔的半无限空间区域映射为圆环域,然后把这个区域内的解析函数展开成Laurent级数的形式,利用Muskhelishvili的复变函数理论和最小二乘法来确定解析函数的各项系数,从而求得浅埋隧道围岩压力的半数值、半解析解,最后通过算例给出了围岩应力的分布情况。计算结果表明,该方法计算精度高、计算量小,具有应用价值。